Systems and Methods to Increase the Durability of Carbonate Reservoir Acidizing

ABSTRACT

Systems and methods for predicting and optimizing the effects of acidizing treatment of carbonate rock are disclosed. The disclosed methods predict the conflicting effects of increased production (i.e., wormhole creation) and reduced rock compressive strength due to acid rock reactions. The mechanical stability of stimulated wellbores, such as horizontal wellbores, can be determined under different acidizing conditions, such as acid type and volume. The acidizing conditions can be optimized to maximize short and long-term production.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. Non-Provisional applicationSer. No. 17/287,014, filed Apr. 20, 2021, which is a 371 National PhaseApplication of PCT (International) Application Serial No.PCT/US2018/057774, filed Oct. 26, 2018. Both of these applications areincorporated herein by reference, and priority is claimed to both.

FIELD OF THE TECHNOLOGY

The present application relates to techniques for carbonate reservoiracidizing.

BACKGROUND

Carbonate reservoir acidizing is a widely used technique to stimulatewells, whereby an acid solution is injected into the formation to inducebranches of open channels by dissolving minerals and by-passing damageto the near-well formation during drilling and well completionprocesses. The objective in this process is to reduce the resistance tothe flow of reservoir fluids due from a naturally tight formation, oreven to reduce the resistance to flow of reservoir fluids due to damage.The efficiency of such a process depends on the type of acid used,injection conditions, structure of the medium, fluid to solid masstransfer, reaction rates, etc. While dissolution increases the rockporosity and permeability, the relative increase in the production for agiven amount of acid is observed to be a strong function of theinjection conditions.

SUMMARY

Embodiments disclosed herein provide methods of acidizing a formationtraversed by a wellbore. According to some embodiments, the methodcomprises determining an optimized acidizing fluid. According to someembodiments, the method comprises providing the optimized acidizingfluid to the formation. According to some embodiments, determining theoptimized acidizing fluid comprises providing a set of acidizingparameters. According to some embodiments, determining the optimizedacidizing fluid comprises determining a distribution of reactive andnon-reactive fluids along the wellbore during acidizing based on the setof acidizing treatment parameters. Some embodiments comprise determininga dissolution of the formation within a region of the wellbore based onthe determined distribution of reactive and non-reactive fluids. Someembodiments comprise determining one or more mechanical parameters ofthe formation within the region of the wellbore based on the determineddissolution. Some embodiments comprise predicting damage to theformation during hydrocarbon production based on the determinedmechanical parameters. Some embodiments comprise predicting an amount ofhydrocarbon produced during hydrocarbon production based, at least inpart, on the predicted damage. Some embodiments comprise adjusting theset of acidizing parameters to maximize the predicted amount ofhydrocarbon produced during hydrocarbon production. According to someembodiments, the set of acidizing parameters comprise one or moreparameters selected from the group consisting of an acid strength, anacid concentration, an acid volume, and an acid injection rate.According to some embodiments, determining a distribution of reactiveand non-reactive fluids along the wellbore comprises determining awellbore flow model of the formation based on an initialcharacterization of the formation and one or more multi-physics and/ormultiscale simulations of the formation. According to some embodiments,the initial characterization of the formation comprises one or moreformation parameters selected from the group consisting of a type ofhydrocarbon fluid in the formation, a configuration of the wellbore, astatic fluid inside the wellbore, a distribution of rock propertiesalong the wellbore, a completion type, and a production enhancement.According to some embodiments, determining a distribution of reactiveand non-reactive fluids along the wellbore during the acidizingtreatment comprises determining wormhole penetration into the formation.According to some embodiments, determining a dissolution of theformation within a region of the wellbore comprises determining aporosity profile of the formation within the region of the wellbore.According to some embodiments, determining one or more mechanicalparameters of the formation comprises determining one or more mechanicalparameters selected from the group consisting of Young's modulus, bulkmodulus, shear modulus, cohesion, internal friction angle, andpore-collapse pressure. According to some embodiments, predicting damageto the formation during hydrocarbon production comprises determining oneor more of shear failure and compressive failure within the formation.According to some embodiments, the one or more of shear failure andcompressive failure within the formation is determined based on porosityof the formation. According to some embodiments, predicting damage tothe formation during hydrocarbon production comprises determining one ormore of shear failure and compressive failure that extends beyond awormhole penetration radius from the wellbore.

Further aspects of the disclosure provide methods of optimizing anacidizing treatment of a formation traversed by a wellbore. According tosome embodiments, the method comprises, for an initial set of acidizingparameters, determining a distribution of reactive fluid along thewellbore. According to some embodiments, the method comprises using afinite element model to determine porosity evolution along the wellborebased on the determined distribution of reactive fluid. According tosome embodiments, the method comprises determining one or moremechanical parameters of the formation based on the determined porosityevolution. According to some embodiments, the method comprisesdetermining a damage radius along the wellbore based on the one or moremechanical properties. According to some embodiments, the methodcomprises predicting an amount of hydrocarbon produced from theformation during hydrocarbon production based on the determined damageradius. According to some embodiments, the method comprises adjustingthe initial set of acidizing parameters to maximize the predicted amountof hydrocarbon produced during hydrocarbon production. According to someembodiments, the mechanical parameters of the formation comprise one ormore parameters selected from the group consisting of Young's modulus,bulk modulus, shear modulus, cohesion, internal friction angle, andpore-collapse pressure. According to some embodiments, the methodfurther comprises determining an extent of wormhole penetration into thewellbore. According to some embodiments, the method further comprisescomparing the extent of wormhole penetration into the wellbore to thedetermined damage radius along the wellbore.

Further aspects of the disclosure provide embodiments of anon-transitory computer readable medium having instructions storedtherein, which when executed by a computer cause the computer to performvarious operations. According to some embodiments, the operationscomprise, for an initial set of acidizing parameters, determining adistribution of reactive fluid along the wellbore. According to someembodiments, the operations comprise determining a dissolution of theformation within a region of the wellbore based on the determineddistribution of reactive and non-reactive fluids. According to someembodiments, the operations comprise determining one or more mechanicalparameters of the formation within the region of the wellbore based onthe determined dissolution. According to some embodiments, theoperations comprise predicting damage to the formation duringhydrocarbon production based on the determined mechanical parameters.According to some embodiments, the operations comprise predicting anamount of hydrocarbon produced during hydrocarbon production based, atleast in part, on the predicted damage. According to some embodiments,the operations comprise adjusting the set of acidizing treatmentparameter to maximize the predicted amount of hydrocarbon producedduring hydrocarbon production. According to some embodiments, theinitial set of acidizing parameters comprise one or more parametersselected from the group consisting of an acid strength, an acidconcentration, an acid volume, and an acid injection rate. According tosome embodiments, determining a distribution of reactive andnon-reactive fluids along the wellbore during the acidizing treatmentcomprises determining wormhole penetration into the formation. Accordingto some embodiments, determining a dissolution of the formation within aregion of the wellbore comprises determining a porosity profile of theformation within the region of the wellbore. According to someembodiments, determining one or more mechanical parameters of theformation comprises determining one or more mechanical parametersselected from the group consisting of Young's modulus, bulk modulus,shear modulus, cohesion, internal friction angle, and pore-collapsepressure. According to some embodiments, predicting damage to theformation during hydrocarbon production comprises determining one ormore of shear failure and compressive failure that extends beyond awormhole penetration radius from the wellbore.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic representation of a cross-section of a wellboreand formation.

FIG. 2 shows mechanical, chemical, thermal and hydrological processesoccurring as acid is injected into carbonate rocks.

FIG. 3 shows a workflow for optimizing acidizing parameters.

FIG. 4 shows forces acting on a section of a wellbore.

FIGS. 5A, 5B, and 5C show porosity, permeability, and stresses,respectively, on a section of a wellbore.

FIGS. 6A, 6B, and 6C show wellhead pressure, wormhole penetration, andwellbore skin, respectively, along a section of a wellbore.

DETAILED DESCRIPTION

As acid is injected into carbonate rock, the local pressure increases,local rock temperature changes, and the injected acid dissolves the rockand increases the local porosity (i.e. by different dissolutionpatterns). FIG. 1 illustrates how the rock porosity is changeddifferently at different radial distances from the wellbore at wormholedissolution pattern.

One of the sources of damage in stimulated carbonate rock is the loss ofnear wellbore formation compressive strength, which may result in casingcollapse under extreme conditions. The differential rock weakening atdifferent radial distance from the wellbore can cause rock to becompacted during production time and can counterbalance the stimulationeffect. Field studies have shown that the failure of acidized rock maycause damage and thereby negatively impact production. Thus, whileacidizing can be effective at stimulating production, particularly inthe near term following treatment, production over extended periods oftime can be reduced due collapse because of failure in the compressivestrength of the formation.

In the instant disclosure, an integrated laboratory measurements andanalysis/design workflow is presented to simulate acid placement incarbonate reservoirs and model stimulated rock behavior during theproductive lifetime of the well. Embodiments of the workflow can beutilized to design a unique approach to carbonate acidizing and optimizethe strategy to maximize hydrocarbon recovery. Aspects of the workflowdescribed herein account for the realization that wormholes inducedduring acidizing may eventually collapse during production, which cannegatively impact production. Thus, the disclosed workflow seeks tooptimize acidizing conditions to maximize hydrocarbon recovery bybalancing the increased short-term production arising because of inducedwormholes and increased hydrocarbon flow against the instability of theformation arising because of rock dissolution.

Aspects of the disclosed methods and workflow allow an operator orservice provider to select and optimize acidizing parameters to maximizelong-term hydrocarbon production following acidizing treatment. Forexample, a service provider may select a particular acid and/or acidconcentration that maximizes hydrocarbon production. Examples of acidsthat may be chosen include, but or not limited to, hydrochloric acid(HCl), acetic acid, hydrofluoric acid (HF), formic acid, and the like.The methods also allow a service provider to select an optimized volumeof acidizing fluid to use and an optimized placement for the acidizingfluid, optimized injection rates, flush rates and the like.

FIG. 2 illustrates some of the interrelated mechanical, chemical,thermal and hydrological processes occurring simultaneously as acid isinjected into carbonate rocks. Such processes include:

101 Stress state variations due to acid injection;

102 Pore pressure alterations due to effective stress changes;

103 Pore pressure changes due to temperature variation;

104 Temperature changes due to fluid flow;

105 Thermal induced stress;

106 Mechanical energy conversion into thermal energy;

107 Pore pressure variation due to acid concentration changes;

108 Acid concentration changes due to fluid flow;

109 Temperature changes due to chemical reaction;

110 Chemical activity variations due to temperature changes;

111 Chemical potential changes due to mechanical energy; and

112 Damage and elasticity factor changes due to chemical reaction.

Including all of these processes and factors into a stimulation analysisand design can be impractical. The disclosed methods simplify therelationships between various of these factors to provide an integratedapproach to analyze carbonate acidizing and to maximize hydrocarbonrecovery by optimizing stimulation design. Aspects of the disclosedmethods (1) determine mechanical damage to acidized rock during posttreatment conditions, and (2) optimize initial stimulation design, (i.e.by changing type of acid, acid concentration, type of diversion, etc.).The optimized acidizing package minimizes the induced mechanical damageand at the same time maximize the long-term production.

Embodiments of the systems and methods described herein include verifiedsimulations and validated and calibrated engineering solutions toeffectively design carbonate reservoir stimulation. FIG. 3 illustratesan embodiment of a workflow 300 as described herein.

According to one embodiment, the workflow 300 utilizes an initialgeomechanical and reservoir characterization of the formation 302. Thesuccess of acidizing and the increase in production is tied to the rightselection of stimulation fluids/technology based on the type offormation, reservoir heterogeneity (i.e. permeability, porosity,pressure, or temperature contrast along the productive length of well),and the length of treatment. Embodiments of the optimized stimulationplan relies on field data, drilling data, completion data, corecharacterization, reliable logging data, core-log correlation, andexpected production after stimulation. Key data that included in theinitial geomechanical and reservoir characterization 302 of theformation can include:

-   -   Type of hydrocarbon fluid, (i.e. black oil, volatile oil,        retrograde gas, wet gas, or dry gas);    -   Wellbore schematics, (i.e. a wellbore configuration such as a        drilling survey, casing and tubing information, etc.);    -   Treatment conducting tubing properties, (i.e. production tubing,        wellbore annulus, coiled tubing, etc.);    -   Static fluid inside the wellbore, (i.e. static column of oil,        emulsion, water, or completion fluid);    -   Wellbore drainage geometry and its driving mechanism;    -   Distribution of rock properties along the wellbore at the time        of stimulation. This can be classified into three major        categories:        -   Core scale reservoir/geological properties: rock mineralogy            along the well, rock density, pore pressure, temperature,            permeability, porosity, pore fluid viscosity, total            compressibility, estimated damage radius, and damage rock            permeability/porosity);        -   Pore scale reservoir properties: pore structure, interfacial            area, areal average of pore radius, and permeability            evolution law by increasing porosity, asymptotic Sherwood            number for pore structure, and pore length to pore diameter            ratio; and        -   Geomechanical properties profile along the wellbore: in-situ            stress, critical porosity, Young's modulus, Poisson's ratio,            unconfined compressive strength (UCS), internal friction            angle, and pore collapse pressure;    -   Completion type along the wellbore, (i.e. open-hole completion,        perforated casing, slotted liner, or combination); and    -   Production enhancement target for the stimulation.

According to some embodiments, the workflow 300 uses multi-physics andmulti-scale simulations 304 to model (1) reactive fluid flow inanisotropic heterogeneous carbonate rock during acidizing and (2)acidized rock behavior during production based on coupledflow-geomechanics simulation. The characteristics of carbonate rockdissolution are affected by acid mass transfer at Darcy scale andacid-carbonate reaction at the pore scale. The modeling of reactivefluid flow in anisotropic heterogeneous carbonate rock during acidizingconsiders acid mass dispersion at Darcy-scale and acid-carbonatereaction at pore-scale to investigate 3D rock dissolution configurationnear a wellbore. The multi-physics-based simulation improves theprediction of carbonate dissolution pattern by studying the influence ofvarious rock environments and operational parameters like injectionrate. For example, the simulation can predict dissolution patterns, suchas face dissolution, conical wormhole formation, wormhole, ramifiedwormhole, and uniform dissolution regimes as a function of injectionrate.

The coupled flow geomechanics simulation provides knowledge ofdissolution phenomena and gain insights regarding to carbonate rockbehavior during acid injection. It also informs the impact of thedissolution on long-term well performance.

Laboratory experiments 306 can be used to verify the simulations. Thepurpose of the simulation verification is to confirm that the simulationprediction is working as intended. Laboratory experiments can be used toidentify and eliminate implementation errors within the code (softwarequality assurance) and to verify the correctness of the numericalalgorithms that are implemented in the code (numerical algorithmverification). Once the modeling of reactive fluid flow in anisotropicheterogeneous carbonate rock during acidizing is verified, the coupledflow-geomechanics simulation can be examined by comparing the stabilityand productivity characteristics of wormholes predicted by the coupledsimulation to hydrostatic loading experiments. During second simulationverification, core-plug samples can be acidized and subsequently mountedin a compaction cell and hydrostatically loaded. According to someembodiments, three experimental measurements can be compared withcoupled flow-geomechanics simulation during hydrostatic loading: (1)pressure drop between start and end of core plug, (2) volumetric strain,and (3) compressibility.

As strong or weak acid injected into carbonate rocks, geochemicalreactions between the acid and the host formation occurs. This leads torock dissolution and modifications of the rock flow and geomechanicalproperties. Dissolution patterns in core plugs can be formed by acidflow. The acid-flooded samples can be mounted in a compaction cell andhydrostatically loaded beyond pore-collapse pressure. Dissolutionpattern collapse and pore collapse can be checked in stress-straincurves. Dissolution pattern collapse can also be checked by CT images ofcompacted samples. Mechanical and physical properties of the samples canbe predefined for any acid-flooding test. Exemplary properties includedensity, permeability, porosity, ultimate compressive strength, modulusof elasticity, Poisson's ratio, cohesion, friction angle, and porecollapse pressure. Chemical composition of the rock can also beidentified, for example, using x-ray fluorescence (XRF) or scanningelectron microscopy (SEM) elemental analysis.

The tests can be carried out on core plugs, for example, 1.5″-diameterplugs. In a first step, acid is flowed through the sample, resulting indissolution formation, and the acid flow is stopped after itbreakthrough. In a second step, the samples can be mounted in acompaction cell and hydrostatically loaded beyond pore-collapsepressure. CT scans of the samples can be taken before and after thecompaction tests. The samples can be mounted in a Hassler cell thatallows for fluid flow through the sample and application of confiningstress.

The results of the multi-physics and multi-scale simulation(s) andcoupled flow-geomechanics simulation(s) are incorporated into aquantitative integrated engineering solution 308 that describes theequivalent flow enhancement by acidizing. The quantitative engineeringsolution can determine the type of dissolution pattern and the extent ofthe pattern, i.e., its “equivalent size”. The dependence of reservoirproperties on equivalent size can be studied based on validatedmulti-physics simulation. The validated simulation results can beinterpreted and employed to derive an engineering correlation thatrelates equivalent size and dissolution pattern to reservoir propertiesand operational parameters.

The quantitative integrated engineering solution 308 can be utilized inthe disclosed integrated approach to analyze carbonate acidizing andsubsequently maximize the hydrocarbon recovery by optimizing stimulationdesign.

The quantitative integrated engineering solution 308 comprises a coupledwellbore flow model 310, which considers fluid interfaces, transientreservoir inflow, and dynamic skin evolution due to rock dissolution.The coupled wellbore flow model 310 simulates the dynamic change ofinjectivity as acid moves inside the wellbore and contacts the rock(i.e. acid contacts top reservoir first and increase the injectivitybefore it reaches to lower reservoir). Within the coupled wellbore flowmodel, the wellbore and reservoir can be divided into many segments. Foreach segment, the model considers the interactions between differentrocks at different locations along the well during the stimulation. Forexample, within a segment, the model can consider different fluids,inflow rates, outflow rates and leak-off into the reservoir, andreservoir characteristics at each segment.

According to some embodiments, the coupled wellbore flow model 310integrates a wellbore flow module, which calculates hydrostatic pressuredrop, as:

$\begin{matrix}{\frac{\partial{Pw}}{\partial x} = {{\rho \times {\mathcal{g}} \times {dz}} \mp \frac{2f_{f}u^{2}\rho}{D}}} & (1)\end{matrix}$

where x is wellbore direction, P_(w) is wellbore pressure, p is fluiddensity, g is the gravitational constant, dz is a change in elevation,f_(f) is Fanning friction factor, u is fluid velocity, and D ishydraulic diameter of wellbore. The positive or negative sign in theequation can be defined based on the direction of flow inside wellbore.

The rate of flow rate change inside wellbore can be represented bymaterial balance equation:

$\begin{matrix}{\frac{\partial q_{w}}{\partial x} = {- q_{R}}} & (2)\end{matrix}$

where q_(w) is the flow rate inside the wellbore, and q_(R) is the fluidinjection rate into reservoir.

The fluid interfaces in the model can be tracked at each time based onthe material balance. It is assumed that the total time of stimulationis divided into small time steps and the condition of wellbore andreservoir is updated after each time step. It is also assumed thatwellbore pressure and flow rate are constant during each time step.

The wellbore pressure dynamics for the duration of stimulation can beconsidered. The superposition is applied to account for wellborepressure variation:

$\begin{matrix}{{- \frac{2\pi{kl}}{\mu}( {P_{R} - P_{W}} )} = {{\sum_{j = 1}^{n}{\Delta{q_{j}\lbrack {p_{D}( {t_{n} - t_{j - 1}} )} \rbrack}}} + {q_{n}s_{n}}}} & (3)\end{matrix}$

where k is the permeability of reservoir rock, l is the length ofwellbore segment, μ is the viscosity of reservoir fluid, P_(R) isreservoir pressure at each element at current condition, P_(W) iswellbore pressure at time t_(n), Δq_(j)=q_(j)−q_(j−1), t_(n) is nth timestep, q_(n) is inflow into reservoir at time t_(n), s_(n) is the localskin factor that changes continuously when acid reacts with the rock andalso when the particulate diversion system hits the formation, and p_(D)is dimensionless pressure function that can be defined based on thedirection of well (vertical or horizontal), reservoir shape, and thetime of injection.

Note that regardless of wellbore direction (i.e. vertical, horizontal,or deviated wellbore), it may be assumed that the fluid is invaded intoreservoir perpendicular to the wellbore direction. Equations (1) and (3)coupled together thru wellbore material balance equation (2) and theresultant algebraic equation is solved with appropriate initial andboundary conditions.

According to some embodiments, the coupled wellbore flow model 310 caninclude a module that predicts wormhole penetration under downholeconditions. According to some embodiments, the model is based ondiffusion being the limiting mechanism for acid transport and does notconsider the role of fluid loss. Examples of methods for predictingwormhole penetration are described in the literature. See, e.g.,Daccord, G., Lenormand, R., & Liétard, O. (1993) Chemical dissolution ofa porous medium by a reactive fluid—I. Model for the “wormholing”phenomenon, Chemical Engineering Science, 48(1), 169-178; and Daccord,G., Touboul, E., & Lenormand, R. (1989) Carbonate Acidizing: Toward aQuantitative Model of the Wormholing Phenomenon SPE ProductionEngineering, 4(01), 63-68. This model may overestimate the distance ofwormhole penetration, but it can be modified thru comparison betweencore studies and field results, as described below.

According to some embodiments, the coupled wellbore flow model 310 caninclude a skin model that accounts for completion skin (perforationcompletion and slotted liner completion), near wellbore condition skin(i.e. permeability change or fluid viscosity change), and thecombination of these skins with wormhole penetration dynamics.

The components of the coupled wellbore flow model 310 are combined andthe resultant equations solved by considering a discretized form of thewellbore. The model provides the following outputs: (1) a prediction ofwellhead and bottom hole pressure during stimulation; (2) dynamics ofwellbore skin improvement during stimulation; and (3) a distributionprofile of different fluid along reservoir. The distributions ofreactive and non-reactive fluids along the wellbore at times duringstimulation can be considered as the input for rock dissolution model312, as shown in FIG. 3 .

Referring again to FIG. 3 , the integrated engineering solution 308 cancomprise a rock dissolution model 312 that simulates rock dissolutionalong the wellbore. As acid reacts with carbonate rock, it dissolvessoluble minerals and increases the porosity of the formation. There aredifferent types of dissolution patterns based on live acid injectioninto rock. At low injection rates, all the soluble material may bedissolved (face dissolution) and the rock permeability rises toinfinity. At high injection rates, rock porosity is increasedhomogeneously (uniform dissolution) and permeability enhanced.Intermediate injection rates may create long infinite conductivechannels called wormholes. The formation of wormholes is also highlydependent on rock heterogeneity, pore structure, molecular diffusion,and reaction rate. Wormhole dissolution patterns may be considered themost optimized pattern since it is the most extensive dissolution withthe lowest volume of acid. However, many operational and environmentalconditions may prevent the formation of wormholes. This necessitates aphysics-based model capable of describing different rock dissolutionpatterns and acid transport in a carbonate rocks.

Carbonate dissolution occurs when live acid transports in the rock anddissolves the rock at the surface of pore structures. Acid transporttakes place at the Darcy scale and rock dissolution occurs at the porescale. Therefore, a two-scale model best describes the phenomenon.Two-scale models based on Partial Differential Equation (PDE) can besolved numerically in four dimensions (4D, i.e., Space and time) todetermine the final rock dissolution pattern. However, with currentcomputational power it may be impractical to apply the 4D numericalsimulation at every measured depth along the wellbore.

The coupled wellbore, reservoir, and wormhole models provided hereinprovide the variations of reactive and non-reactive fluid volumesqueezed into every measured depth along the stimulation timeframe. Itmay be assumed here that the acid radially flows and dissolves the rockand that there is no tangential acid flow. Taking this assumption intoaccount, the developed PDEs can be non-dimensionalized in thecylindrical coordinates system using following dimensionless variables:

$\begin{matrix}{{\xi = \frac{r}{r_{e}}},{\alpha_{w} = \frac{r_{0}}{r_{e}}},{u_{r} = \frac{u}{u_{0}}},{t^{\prime} = \frac{t}{r_{e}/u_{0}}},{\zeta_{p} = \frac{r_{p}}{r_{p0}}},{A_{v} = \frac{a_{v}}{a_{0}}},} \\{{\kappa = \frac{K}{K_{0}}},{c_{f} = \frac{c_{f}}{c_{0}}},{\eta = \frac{2r_{p0}}{r_{e}}},{\phi^{2} = \frac{2k_{s}r_{p0}}{D_{m}}},{D_{a} = \frac{k_{s}a_{0}r_{e}}{u_{0}}}} \\{{N_{ac} = \frac{\alpha C_{0}}{\rho_{s}}},{P_{e} = \frac{u_{0}r_{e}}{D_{m}}},{\Phi^{2} = \frac{k_{s}a_{0}r_{e}^{2}}{D_{m}}}}\end{matrix}$

where r is the radius in cylindrical coordinate, r_(e) is the externalradius of the model that fluid radial velocity is negligible, r₀ iswellbore radius, u is radial velocity as a function of radius and time,u₀ is the inlet radial velocity of reactive or non-reactive fluid intoevery reservoir section along the wellbore as a function of time (asprovided from coupled wellbore, reservoir, and wormhole models), t istime, r_(p) is pore radius as a function of radius and time, r_(p0) isthe initial pore radius, a_(v) is interfacial pore area as a function ofradius and time, a₀ is initial interfacial pore area, K is permeabilityas a as a function of radius and time, K₀ is initial rock permeability,C_(f) is concentration of acid as a function of radius and time, C₀ isinlet acid concentration as a function of time (as provided from thecoupled wellbore, reservoir, and wormhole models), D_(m) is theeffective molecular diffusivity of the reactive fluid, k_(s) is thesurface reaction rate constant with the velocity unit, a is aciddissolving power, which is defined as the grams of solid dissolved permole of acid. The introduced dimensionless parameters are pore-scaleThiele number ϕ², the Damkohler number, the acid capacity number N_(ac),the radial Peclet number P_(e), and macroscopic Thiele number Φ².

The one-dimensional dimensionless equations in cylindrical coordinatesare:

$\begin{matrix}{\frac{\partial\varepsilon}{\partial t^{\prime}} = {\frac{1}{\xi}{\frac{\partial}{\partial\xi}( {\xi u_{r}} )}}} & (4)\end{matrix}$ $\begin{matrix}{{\frac{\partial( {\varepsilon c_{f}} )}{\partial t^{\prime}} + {\frac{1}{\xi}{\frac{\partial}{\partial\xi}( {\xi u_{r}c_{f}} )}}} = {{\frac{1}{\xi}{\frac{\partial}{\partial\xi}\lbrack {\xi\{ {\frac{\alpha_{os}\varepsilon D_{a}}{\Phi^{2}} + {\lambda_{r}{❘u_{r}❘}\zeta_{p}\eta}} \}\frac{\partial c_{f}}{\partial\xi}} \rbrack}} - \frac{D_{a}A_{v}c_{f}}{( {1 + \frac{\Phi^{2}\zeta p}{Sh}} )}}} & (5)\end{matrix}$ $\begin{matrix}{\frac{\partial\varepsilon}{\partial t} = \frac{D_{a}N_{ac}A_{v}c_{f}}{( {1 + \frac{\Phi^{2}\zeta p}{Sh}} )}} & (6)\end{matrix}$

Equation (4) describes the dimensionless continuity condition andrelates the increase in rock local porosity to the change in the localvelocity. Equation (5) presents the dimensionless mass balance for theacid species. The first and second terms on the left-hand side are theaccumulation and convection terms, respectively, while the first term onthe right-hand side represents dispersion of acid species. The secondterm on the right-hand side represents the mass transfer of acid speciesfrom the bulk fluid phase to the fluid-solid interface. Equation (6)states that the mass of acid species transported to the fluid-solidinterface is reacted on the surface. In the above equations aos is aconstant that depends on the structure of the porous medium (forexample, tortuosity or connectivity between the pores), Ur is radialvelocity, Sh is Sherewood number, and ϵ is porosity.

According to some embodiments, the corresponding boundary and initialconditions are:

$\begin{matrix}{{{\begin{Bmatrix}{u_{r} = 1.} \\{c_{f} = 1.} \\{\frac{\partial\varepsilon}{\partial\xi} = 0.}\end{Bmatrix}@\xi} = \alpha_{w}},{{\begin{Bmatrix}{u_{r} = 0.} \\{\frac{\partial c_{f}}{\partial\xi} = 0.} \\{\varepsilon = \varepsilon_{0}}\end{Bmatrix}@\xi} = 1},{{\begin{Bmatrix}{{u_{r}(\xi)} = 0.} \\{{c_{f}(\xi)} = 0.} \\{{\varepsilon(\xi)} = \varepsilon_{0}}\end{Bmatrix}@t^{\prime}} = 0.}} & (7)\end{matrix}$

where cε₀ is the initial porosity of the media. The coupled transientPDE equations (4), (5), and (6) with boundary and initial conditions (7)can be solved using Finite Element Method and the porosity and theradial profile of porosity acid concentrations can be calculated. Thefinal porosity profile for each measured depth can be provided to the 3Drock mechanical model 314 which models the weakened rock and calculatesmechanical properties of rock at each radius.

Referring again to FIG. 3 , the embodiments of the integratedengineering solution 308 includes a 3D rock mechanical model 314 thatmodels how the rock is weakened due to acidizing. Rock dissolution (i.e.face dissolution or wormhole propagation, etc.) is a major reason foracidized well production enhancement. However, as acid is injected intowellbore, rock is weakened at different levels and stresses around thewellbore can be changed. The stress changes can neutralize theenhancement of acid injection by breaking and closing induced wormholesand in some instances, may induce casing collapse.

Different dissolution patterns differently affect rock stiffness andstrength. For example, face dissolution only increases wellbore radiusand does not generally considerably alter the remaining rock stiffnessand strength. Induced wormholes and/or uniform dissolution generallyhave the greatest effect on stiffness and strength reduction of rock. Asthe near wellbore rock loses stiffness, it will pass the excess load tothe adjacent rock, which may collapse.

According to some embodiments of the disclosed 3D mechanical modeling,the effective total porosity of acidized rock is assumed to be areliable indicator of rock stiffness and strength parameters. Theeffective total porosity of stimulated rock at each measured depth alongthe wellbore is provided by the 3D rock mechanical model 314. Generally,the mechanical properties of the stimulated rock can be determined usingany method known in the art, which methods have varying degrees ofcomplexity. Embodiments of the disclosed 3D rock mechanical model usesanalytical models that allow one to estimate elastic and failureproperties of carbonates from porosity. Examples of rock mechanicalmodeling are described in the literature. See, for example, Bemer, E.,Vincké, O., & Longuemare, P. (2004) Geomechanical Log Deduced fromPorosity and Mineralogical Content, Oil & Gas Science andTechnology-Rev. IFP, 59(4), 405-426; Bauer, A., Walle, L. E.,Stenebraten, J., & Papamichos, E. (2013) Impact of Acidizing-InducedWormholes in Chalk on Rock Strength, presented at the 47th U.S. RockMechanics/Geomechanics Symposium, San Francisco, Calif., Dormieux, L.,Jeannin, L., Bemer, E., Le, T. H., & Sanahuja, J. (2010) Micromechanicalmodels of the strength of a sandstone, International Journal forNumerical and Analytical Methods in Geomechanics, 34(3), 249-271doi:10.1002/nag.804; and Nguyen, M. T., Bemer, E., & Dormieux, L. (2011)Micromechanical Modeling of Carbonate Geomechanical Properties EvolutionDuring Acid Gas Injection, presented at the 45th U.S. RockMechanics/Geomechanics Symposium, San Francisco, Calif. The rockmechanical properties (i.e., mechanical parameters) include bulkmodulus, shear modulus, cohesion, internal friction angle, andpore-collapse pressure. The following relations provides reasonableestimation of elastic and plastic properties for limestone.

$\begin{matrix}{{{K(\varepsilon)} = \frac{( {1 - \varepsilon} ) \times K_{S}}{1 - \varepsilon + {\varepsilon\frac{K_{S}}{K_{C}}}}},{K_{S} = {72.6{GPa}}},{\frac{K_{C}}{K_{S}} = 0.07}} & (8)\end{matrix}$ $\begin{matrix}{{{G(\varepsilon)} = \frac{( {1 - \varepsilon} )G_{S}}{1 - \varepsilon + {\varepsilon\frac{G_{S}}{G_{C}}}}},{G_{S} = {31.6{GPa}}},{\frac{G_{C}}{G_{S}} = 012}} & (9)\end{matrix}$ $\begin{matrix}{{{c^{\prime}(\varepsilon)} = {c_{0} \times {\exp( {- \alpha_{c} \times \varepsilon} )}}},{\varepsilon = {0/0}},{c_{0} = {40.3{MPa}}},{\alpha_{c} = 0.054}} & (10)\end{matrix}$ $\begin{matrix}{{{\varphi^{\prime}(\varepsilon)} = {{\alpha_{f}\varepsilon} + \beta_{f}}},{\varepsilon = {0/0}},{\alpha_{f} = {- 0.893}},{\beta_{f} = 49.}} & (11)\end{matrix}$ $\begin{matrix}{{{p^{*}(\varepsilon)} = {p_{0}{\exp( {- \alpha_{p} \times \varepsilon} )}}},{\varepsilon = {0/0}},{p_{0} = {601.6{MPa}}},{\alpha_{p} = 0.083}} & (12)\end{matrix}$

where K(ϵ) is bulk modulus as a function of porosity, G(ϵ) is shearmodulus as a function of porosity, ć(ϵ) is cohesion as a function ofporosity, φ(ϵ) is internal friction angle as a function of porosity,p*(ϵ) is pore collapse pressure as a function of porosity. The presentedrock physics model is an example of a limestone weakening modelparameterized by final porosity of stimulated rock. Such a model fordifferent types of rock is utilized in the integrated approach toestimate rock mechanical properties (elastic and plastic) for differentlevels of enhanced porosity. The 3D rock mechanical model 314 providesmechanical properties of the different concentric rock rings about thewellbore (as shown in FIG. 1 ), which are incorporated into thegeomechanical stress analysis discussed below.

Referring again to FIG. 3 , the integrated engineering solution 308includes geomechanical model 316 of the wellbore for determininggeomechanical factors that can lead to collapse within the rock layerswithin the acidized region of the wellbore over the productive life ofthe reservoir. As mentioned above, compaction of the stimulated rock canoccur during production and neutralize the effects of acid stimulation.Therefore, the disclosed workflow analyzes the wellbore under possibledrawdown scenarios to minimize undesired mechanical damage as stimulatedrock is compacted and permeability is decreased.

The geomechanical model 316 provides an analytical stress analysis topredict the stress distribution in the wellbore, considering concentricrings of different mechanical properties, casing, cement, and differentrings of weaken rock with different mechanical properties that aredefined in a rock physical engine. Examples of geomechanical modelingtechniques are described in the literature, for example, in Jo, H.(2008) Mechanical Behaviour of Concentric and Eccentric Casing, Cement,and Formation Using Analytical and Numerical Methods. (PhD), TheUniversity of Texas at Austin and Jo, H., & Gray, K. E. (2010)Mechanical Behavior of Concentric Casing, Cement, And Formation UsingAnalytical And Numerical Methods, presented at the 44th U.S. RockMechanics Symposium, Salt Lake City, Utah. The geomechanical model 316can predict time dependent stresses and consequently shear-enhancedcompaction and/or compaction that might be generated by drawdownvariations inside the wellbore during production. In other words, thegeomechanical model 316 predicts formation damage under various drawdown(i.e., production) conditions.

The determination of acidized rock behavior during productionnecessitates a complete picture of after-stimulation condition.According to some embodiments, the followings are defined for completewellbore stress analysis after acidizing:

-   -   Current state of stress and pore pressure inside of the        reservoir; which includes both magnitude and direction of        principal stresses,    -   Level of near-wellbore rock weakening due to stimulation at        different section along the completed length of a well,    -   Failure and mechanical properties of different weakened rock        near the wellbore, and Stress distribution near the wellbore and        prediction of any brittle or ductile failure.

Methods for predicting stress changes range from relatively simpleanalytical solutions to numerical modeling. According to someembodiments, analytical solutions are employed. Utilized solutions arebased on the theories of inclusions and inhomogeneities and have theadvantage of being easy to implement in the integrated framework due totheir rapid computation times. See, e.g., Soltanzadeh, H., HawkesChristopher, D., and Sharma Jitendra, S. 2007. Poroelastic Model forProduction—and Injection-Induced Stresses in Reservoirs with ElasticProperties Different from the Surrounding Rock. International Journal ofGeomechanics 7 (5): 353-361.

As acid reacts with carbonate rock, it dissolves soluble minerals,increases rock porosity, and induces different types of dissolutionpattern. Relatively, any rock dissolution pattern (i.e. face dissolutionor wormhole propagation) is the major reason for short or long-termproduction enhancement. However, it should be kept in mind that as acidis squeezed into the rock, the porosity is increased, and the rock isweakened at different levels as shown in FIG. 1 . Accordingly, thestress concentration around the wellbore changes which might result ininduced wormhole closure and subsequently neutralizing the effect ofacid injection and in worst condition induces casing collapse.

According to some embodiments, it is assumed that the porosity ofstimulated rock is an indicator of rock stiffness and strengthparameters. The porosity of stimulated rock at each measured depth alongthe wellbore is determined based on the rock dissolution model 312 (FIG.3 ). According to some embodiments, an underlying assumption ofhomogeneous rock porosity and permeability. As mentioned above, the rockdissolution model 312 calculates rock mechanical properties (i.e.,mechanical parameters) including bulk modulus, shear modulus, cohesion,internal friction angle, and pore-collapse pressure.

FIG. 5 shows the geometry used to describe the stimulated wellbore underin-situ condition during post-stimulation production. The wellbore isassumed to have several concentric rings of carbonate rock withdifferent mechanical properties. The outer ring is far enough to assumeintact rock properties and close rings to the wellbore might be casingor cement sheath, if they are part of completion. The vertical stressS_(v), horizontal stresses (S_(H) and S_(h)), wellbore pressure P_(W),and reservoir pressure PR are at current reservoir condition afterstimulation. Moreover, temporal changes of wellbore pressure duringpost- stimulation are also considered.

There are numerous modeling approaches to calculate stress distributionaround the stimulated wellbore. They can be categorized into analyticalelastic/poroelastic solutions, and numerical elastoplastic models.Existing analytical solutions are not 3D and they are based on 2D plainstrain theory. Albeit a wellbore satisfies requirements to apply the 2Dplane strain condition, the axial tectonic stress along the wellboreinduces in-plane stresses and makes the system a 3D problem. Accordingto some embodiments, generalized plain strain theory is used to addressthe 3D characteristics of the problem. The engine can predict timedependent effective stresses generated by drawdown variations inside thewellbore during production.

According to some embodiments, a constitutive model determines theregion in a representative stress space outside of which the rock cannottolerate load and fails. The representative stress state is described byTerzaghi's effective stress state. See Vincke, 0., Boutéca, M. J., Piau,J. M., and Fourmaintraux, D. 1998. Study of the Effective Stress atFailure. In Biot Conference on Poromechanics: 635-639. Since thestimulated rock is continuously under compressive loading, merely shearand compaction failure need be considered, according to someembodiments.

With carbonate rocks, the porosity level mainly controls rock yielding.At low porosity, the effective mean pressure at initial yielding and thegap between the hardening onset and the initial yielding are very high.High effective mean stress triggers shear-enhanced compaction andhardening due to grain crushing and pore collapse. On the other hand,brittle failure occurs at low effective mean stress. Limestone brittlefailure can be mathematically described by Coulomb's law:

$\begin{matrix}{q = {A + {B \times p^{\prime}}}} & (14)\end{matrix}$ $\begin{matrix}{A = \frac{6 \times c \times \cos\varphi}{3 - {\sin\varphi}}} & (15)\end{matrix}$ $\begin{matrix}{B = \frac{6 \times \sin\varphi}{3 - {\sin\varphi}}} & (16)\end{matrix}$

where p′is effective mean stress, q is deviatoric stress, c is rockcohesion and φ is rock friction angle. The ductile failure (rockcompaction) is also considered to account for possible collapse of rockafter stimulation. The compaction curve is assumed as a circular capthat limits stress state in addition to brittle failure:

q ² +p′ ²=(p*)²   (17)

where p* is grain crushing and pore collapse pressure. To ensure an easyhandling of the model, embodiments suppose that failure in Limestone ismainly controlled by the rock porosity.

Referring again to FIG. 3 , embodiments of the integrated engineeringsolution 308 includes a production prediction engine 318. During theproduction (pre and post-stimulation phases), the wellbore pressure ischanging with time so transient effects are occurring in the reservoir.Superposition principals can be employed to estimate outflow fromdifferent sections of reservoir. The production including the transienteffect and dynamic skin as a result of geomechanical damage can becalculated as:

$\begin{matrix}{{- \frac{2\pi{kl}}{\mu}( {p_{r} - p_{w}} )} = {{\sum_{j = 1}^{n}{\Delta{q_{j}\lbrack {p_{D}( {t_{n}^{D} - t_{j - 1}^{D}} )} \rbrack}}} + {q_{n}s_{n}}}} & (18)\end{matrix}$

where k is the permeability of reservoir rock, l is the length ofreservoir segment (i.e. completion length of a section of reservoir),p_(r) is reservoir pressure, p_(W) is wellbore pressure, μ is viscosityof reservoir fluid, t_(n) ^(D) is dimensionless time at the nth timestep, p_(D) is dimensionless pressure, q_(n) is production rate at thenth time step, s_(n) is the skin factor at the nth time step, andΔq_(j)=q_(j)−q_(j−1). The variables t_(n) ^(D) and P_(D) can havedifferent expressions for transient, late transient, steady state, orpseudo steady state condition of the reservoir. According to someembodiments it can be assumed that there is no cross-flow between eachsection of the wellbore. Note that the skin of each section of thereservoir can be updated based on the wellbore stress analysis andprediction of weakened rock compaction.

According to some embodiments, the wellbore can be divided into smallelements with “l_(i)” length. The equation (18) can be applied for eachsegment to calculate production rate per each unit length of reservoir.By rearranging the equation (18), we have:

$\begin{matrix}{{{- \frac{2\pi{kl}}{\mu}( {p_{r} - p_{w}} )} = {{\sum_{j = 1}^{n - 1}{\Delta{q_{j}^{U}\lbrack {p_{D}( {t_{n}^{D} - t_{j - 1}^{D}} )} \rbrack}}} - {q_{n - 1}^{U} \times {p_{D}( {t_{n}^{D} - t_{n - 1}^{D}} )}} + {q_{n}^{U} \times \lbrack {{p_{D}( {t_{n}^{D} - t_{n - 1}^{D}} )} + s_{n}} \rbrack}}},{{{where}q_{j}^{U}} = \frac{q_{j}}{I_{1}}}} & (19)\end{matrix}$

Further rearrangement yields:

$\begin{matrix}\begin{matrix}{{q_{n}^{U} = {{- A_{j} \times ( {p_{r} - p_{w}} )} - B_{j}}},{where}} \\{{A_{j} = \frac{2\pi k}{µ \times \lbrack {{p_{D}( {t_{n}^{D} - t_{n - 1}^{D}} )} + s_{n}} \rbrack}},{and}} \\{B_{j} = \frac{{\sum_{j = 1}^{n - 1}{\Delta{q_{j}^{U}\lbrack {p_{D}( {t_{n}^{D} - t_{j - 1}^{D}} )} \rbrack}}} - {q_{n}^{D} \times {p_{D}( {t_{n}^{D} - t_{n - 1}^{D}} )}}}{µ \times \lbrack {{p_{D}( {t_{n}^{D} - t_{n - 1}^{D}} )} + s_{n}} \rbrack}}\end{matrix} & (20)\end{matrix}$

Thus, equation (20) allows calculation of q_(n) ^(U), which is the unitproduction rate at every section of the reservoir, including transienteffect of reservoir and reservoir skin evolution as different section ofreservoir might compact during production.

It will be appreciated that the integrated engineering solution 308allows acidizing parameters to be optimized with a view to maximizingproduction from the formation over time, rather than simply maximizingshort term production, wormhole creation, or the like. As the integratedengineering solution accounts for both increases in production becauseof increases in porosity but also counterbalances the likelihood offormation damage due to mechanical weakening of the formation, theoptimized parameters can maximize the time that acidizing treatment iseffective.

The model first simulates stimulation fluid movement in the wellbore,and couples it with transient reservoir flow. The primary analysisprovides the distribution of reactive fluid along the well. The rockdissolution at each section of reservoir is then analyzed by adiscretizing continuum two-scale model with a finite element method(pore-scale and Darcy scale). This analysis presents porosity evolutionat different locations along the well and at different radial distancesfrom the wellbore center. The mechanical properties (elastic and failureproperties) of rock at different radial distances are then computedbased on the porosity alteration. Then, the developed stress analysisengine simulates the weakened rock under the in-situ stresses andbottom-hole pressure and predicts any possible compressional or shearfailure. The model predicts the amount of hydrocarbon produce over time,considering compressional and/or shear failures. According to someembodiments, hydrocarbon production following acidizing treatment ispredicted over a time period of weeks, months, and or years. Forexample, hydrocarbon production over a 6-month period may be predicted.Alternatively, hydrocarbon production over an 18-month time period maybe predicted.

Some embodiments of subject matter and operations described in thisspecification can be implemented in digital electronic circuitry, or incomputer software, firmware, or hardware, including the structuresdisclosed in this specification and their structural equivalents, or incombinations of one or more of them. Some embodiments of subject matterdescribed in this specification can be implemented as one or morecomputer programs, i.e., one or more modules of computer programinstructions, encoded on computer storage medium for execution by, or tocontrol the operation of, data processing apparatus. Specifically, suchprograms/instructions may be stored on a non-transitorycomputer-readable medium. A computer storage medium can be, or can beincluded in, a computer-readable storage device, a computer-readablestorage substrate, a. random or serial access memory array or device, ora combination of one or more of them. Moreover, while a computer storagemedium is not a propagated signal, a computer storage medium can be asource or destination of computer program instructions encoded in anartificially generated propagated signal. The computer storage mediumcan also be, or be included in, one or more separate physical componentsor media (e.g., multiple CDs, disks, or other storage devices).

The term “data processing apparatus” encompasses all kinds of apparatus,devices, and machines for processing data, including by way of example aprogrammable processor, a computer, a system on a chip, or multipleones, or combinations, of the foregoing. The apparatus can includespecial purpose logic circuitry, e.g., an FPGA (field programmable gatearray) or an ASIC (application specific integrated circuit). Theapparatus can also include, in addition to hardware, code that createsan execution environment for the computer program in question, e.g.,code that constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, a cross-platform runtimeenvironment, a virtual machine, or a combination of one or more of them.The apparatus and execution environment can realize various differentcomputing model infrastructures, such as web services, distributedcomputing and grid computing infrastructures.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, declarative orprocedural languages. A computer program may, but need not, correspondto a file in a file system. A program can be stored in a portion of afile that holds other programs or data (e.g., one or more scripts storedin a markup language document), in a single file dedicated to theprogram in question, or in multiple coordinated files (e,g., files thatstore one or more modules, sub programs, or portions of code). Acomputer program can be deployed to be executed on one computer or onmultiple computers that are located at one site or distributed acrossmultiple sites and interconnected by a communication network.

Some of the processes and logic flows described in this specificationcan be performed by one or more programmable processors executing one ormore computer programs to perform actions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andprocessors of any kind of digital computer. Generally, a processor willreceive instructions and data from a read only memory or a random-accessmemory or both. A computer includes a processor for performing actionsin accordance with instructions and one or more memory devices forstoring instructions and data. A computer may also include or beoperatively coupled to receive data from or transfer data to, or both,one or more mass storage devices for storing data, e.g., magnetic,magneto optical disks, or optical disks. However, a computer need nothave such devices. Devices suitable for storing computer programinstructions and data include all forms of non-volatile memory, mediaand memory devices, including by way of example semiconductor memorydevices (e.g., EPROM, EEPROM, flash memory devices, and others),magnetic disks (e.g., internal hard disks, removable disks, and others),magneto optical disks, and CD ROM and DVD-ROM disks. The processor andthe memory can be supplemented by, or incorporated in, special purposelogic circuitry.

To provide for interaction with a user, operations can be implemented ona computer having a display device (e.g., a monitor, or another type ofdisplay device) for displaying information to the user and a keyboardand a pointing device (e.g., a mouse, a trackball, a tablet, a touchsensitive screen, or another type of pointing device) by which the usercan provide input to the computer. Other kinds of devices can be used toprovide for interaction with a user as well; for example, feedbackprovided to the user can be any form of sensory feedback, e.g., visualfeedback, auditory feedback, or tactile feedback; and input from theuser can be received in any form, including acoustic, speech, or tactileinput. In addition, a computer can interact with a user by sendingdocuments to and receiving documents from a device that is used by theuser; for example, by sending web pages to a web browser on a user'sclient device in response to requests received from the web browser.

A client and server are generally remote from each other and typicallyinteract through a communication network. Examples of communicationnetworks include a local area network (“LAN”) and a wide area network(“WAN”), an inter-network (e.g., the Internet), a network comprising asatellite link, and peer-to-peer networks (e.g., ad hoc peer-to-peernetworks). The relationship of client and server arises by virtue ofcomputer programs running on the respective computers and having aclient-server relationship to each other.

In some aspects, some or all of the features described here can becombined or implemented separately in one or more software programs fordetermining acid injection treatment parameters. The software can beimplemented as a computer program product, an installed application, aclient-server application, an Internet application, or any othersuitable type of software.

Example

Aspects of the workflow 300 was applied to optimize the stimulationtreatment of a high temperature horizontal wellbore with around 10500feet measured depth and 250° F. temperature. The effect of stimulationfluid was modeled to optimize short term and long-term production. Twotypes of acid were considered: I) 15% HCl and II) 10% Acetic acid. Aconstant in Daccord model (see Daccord et al. Chemical EngineeringScience 48 (1): 169-178, 1993) was identified for both acids from coreflooding tests and previous field stimulations analysis as known in theart.

The wellbore schematics were as follows: a packer is installed at 8000ft and completed zone is started from 8600 feet to 10475 feet. Thewellbore is completed with cemented casing and perforated with 1 shotper 2 feet. Perforation has 18-inch length, 0° phasing angle, and0.35-inch diameter. Initially the wellbore is filled with oil having aspecific gravity of 0.88 and the viscosity is 1.2 cp at reservoirtemperature. It is assumed that the total compressibility of reservoiris 10-6 psi⁻¹.

Table 1 shows wellbore construction data. It was assumed that bothcasing and tubing has 0.0001 relative roughness. It was assumedreservoir initial pressure was 3500 psi and reservoir temperature was250° F. Porosity, permeability, and stresses (vertical stress S_(V), andhorizontal stresses S_(H) and S_(h)) are shown in FIGS. 5A, 5B, and 5C,respectively. Young's modulus, Poisson's ratio, cohesion, frictionangle, and pore collapse pressure for each section of reservoir alongwellbore were estimated based on initial porosity as known in the art(see Bemer et al., Oil & Gas Science and Technology-Rev. IFP 59 (4):405-426, 2004). It was estimated that the damage radius along thewellbore is 18 inches deep with 10 times permeability reduction. Theresultant skin was approximately 38.

TABLE 1 Wellbore Construction Data. Well Casing Tubing Tubing MD RadiusID Casing ID OD Tubing (ft) (in) (in) Roughness (in) (in) Roughness 330312.25 9.625 0.0001 2.548 3.5 0.0001 5165 8.75 6 0.0001 2.548 3.5 0.00018300 8.75 6 0.0001 2.259 2.875 0.0001 8393 8.75 6 0.0001 0 0 0 105008.75 6 0.0001 0 0 0

The initial stimulation package considered included 300 bbl. ofpre-flush, 1850 bbl. of 15% HCl acid (around 1 bbl. per each feet ofcompletion), and 500 bbl. of post flush. The injection rate and totalvolume of injected acid were first optimized without considering therock behavior during production. It was assumed that the horizontalwellbore was fully penetrated in the middle of a box-shaped drainagearea with 5000 feet length and 1000 feet height. A Babu and Odeh modelwas used for the wellbore production prediction, as is known in the art(see Babu, et al., Productivity of a Horizontal Well. SPE ReservoirEngineering 4 (4), 1989).

FIGS. 6A, 6B, and 6C show the wellhead pressure, wormhole penetration,and wellbore skin along the wellbore, respectively, as predicted by thewellbore flow model 310. As shown in FIG. 6A, the wellbore pressure hasthe highest pressure (around 5800 psi) at the beginning of injection.After pre-flush, acid is injected into the wellbore and as soon as acidsqueezed into the formation, the overall wellbore injectivity increasesand wellhead pressure drops. The wellhead pressure decline rate is verysteep at the beginning and then pressure reaches a plateau. This showsthe effect of induced wormhole length. At the initial stage of acidreaction with rock, small penetration of wormholes has significantcontribution to reduce the skin and enhance flow-capacity. However, aswormhole extended further into the reservoir, its effect diminishes. Thebehavior of wellhead pressure is a clear signature of the statedphysical phenomena.

FIGS. 6B and 6C show the final wormhole length and final skin of thewellbore, respectively. As expected, wormhole penetration follows thepermeability and porosity profile. The wormhole penetration along thewellbore varies from 0 feet to 12 feet. In other words, the inducedwormhole extends beyond the damage radius (18 inch) at high permeabilitylocations and is shorter than damage radius at other locations. This isa manifestation of over stimulating some parts of the well and understimulating the other parts. FIG. 6C demonstrates that the skin variesalong the wellbore. This is total skin as a result of remaining damage(toward the shoe of the well), wormhole penetration, and perforationskin. On average, the skin of well decreased from 38 to around 15 whichis 60% reduction.

The wellbore flow model 310 further predicts different fluid penetrationat the end of injection. As with wormhole penetration, the fluidpenetration follows the permeability and porosity profile. The wellboreflow model 310 determined that the stimulation fluid could not reach theend of wellbore and hence, limited stimulation was achieved at thatlocation (i.e. skin remains in an average of 38). The wellbore flowmodel 310 also showed that the penetration of post flush and justifiesthe large volume that was designed to push most of remaining acid insidewellbore into the reservoir and bypass near wellbore region. Asmentioned above, the wellbore flow model 310 provides an injectionprofile along the wellbore to the rock dissolution model 312. At eachtime during the stimulation, specific fluid type (i.e. reactive likeacid or non-reactive like pre-flush) is injected into the rock. Thedynamic effect of different fluid injection into each section of thereservoir is analyzed and porosity enhancement is calculated by usingrock dissolution model 312.

The rock dissolution model calculated a change of porosity along thewellbore at the end of stimulation. The porosity was found to be 0 to0.6, which is the critical porosity of this carbonate rock. As it isexpected, wellbore enlargement is significant at higher permeabilityregion. The porosity variation along the wellbore at each radialdistance was introduced into the rock mechanical model 314 and thegeomechanical model 316 and the geomechanical properties of each sectionwere calculated.

The mechanical model 314 was used to calculate the following mechanicalproperties along the wellbore: the Young's modulus changes from 0.0 to 7MMpsi, internal friction angle changes from 0.0 to 45, and pore collapsepressure changes from 0.0 to 50000 psi. This data along with otherelastic and failure properties are essential for wellbore stress/failureanalysis during production time. The geomechanical data is provided intothe three-dimensional wellbore stress analyses and the shear failure andcompaction failure of the wellbore at each section and at eachproduction time is assessed. In this case, it is assumed that thewellbore pressure is kept constant at 2000 psi and the stimulation isperformed after 9 months of initial production.

The geomechanical model 316 was used to model the status of nearwellbore rock at different sections of the wellbore after 18 months ofproduction. At different sections of wellbore different levels ofwellbore enlargement occurred. Some sections showed limited wellboreenlargement and also limited rock compaction near the wellbore. In otherwords, the compaction was not extensive enough to counteract the effectof wormholes. Therefore, that compaction would not negatively impact theproduction from those sections. On the hand, other sections exhibitedextensive wellbore enlargement and rock compaction. At those sections,rock compactions extended beyond the wormhole length and neutralized allor part of the stimulation effect. Note that the limited or extensivefailure region at each section is the outcome of acid type and injectionvolume into different section along the well.

Thus, the geomechanical model 316 showed that by using 15% HCl,considerable compaction on high porosity/permeability areas is induced.Those findings were used by the production prediction engine 318 to showthe production prediction before and after the stimulation including thecompaction effect at different sections along the wellbore. The resultsshow that the stimulation after 9 months can bring the production ratefrom 370 bbl/day to around 1000 bbl/day. However, the production ratedeclines to its original trend just 3 months after the stimulation.

To optimize the acid type, the models were rerun using 10% acetic acid.To achieve the same level of wormhole penetration and skin reduction (tomaximize short term production), the volume of acetic acid was increasedto 3600 bbl. (1.9 bbl./ft.). The volume of acetic acid was adjusted togive us same wormhole pattern as HCl in order to have equal (if notbetter) short term initial production. The wellbore flow model anddissolution model showed that the wormhole extension for 10% Acetic acidis slightly shorter than 15% HCl but follows the same pattern anddistribution as 15% HCl case. The modeling showed that the porosityimprovement of the near-wellbore rock is extended less and thereforeweakens the rock less. This is a signature of more competent rock duringproduction that can withstand elevated effective stress duringproduction. Geomechanical modelling proved this behavior. It was shownthat none of the previous section of reservoir that demonstratecompaction with HCl stimulation, experienced excessive compaction withacetic acid stimulation. In other words, the modeling predicted thatutilizing optimized weaker acid at high temperature conditions couldinduce an extensive wormhole with minimum weakening of surrounding rock.The production engine predicted a difference between production ratejust after stimulation of around 85 bbl/day. This difference inproduction pays off the surplus charge for 10% Acetic acid in an orderof 6 months in addition to minimizing induced geomechanical damage.

The model results show that the overall efficiency of the acidstimulation is primarily a function of the stimulation design parameters(acid type and acid volume) and the geomechanical characteristics of thecarbonate rock. It shows that the stimulation effect is stable sinceexcessive weakening increases the rock compaction under near wellboreeffective mean stress. At the elevated mean effective stress, the rockfailure extended away from near-wellbore region and neutralized thestimulation effects.

While the invention herein disclosed has been described in terms ofspecific embodiments and applications thereof, numerous modificationsand variations could be made thereto by those skilled in the art withoutdeparting from the scope of the invention set forth in the claims.

What is claimed is:
 1. A method of stimulating hydrocarbon production ina formation traversed by a wellbore, the method comprising: determiningan optimized stimulation treatment, and providing the optimizedstimulation treatment to the formation, wherein determining theoptimized stimulation treatment comprises: providing a set ofstimulation parameters, determining one or more mechanical parametersindicative of weakening of the formation within a region of the wellborecaused by one or more stimulation parameters of the set of stimulationparameters, predicting a damage to the formation due to formation stressduring hydrocarbon production based on the one or more mechanicalparameters, predicting an amount of hydrocarbon produced duringhydrocarbon production based, at least in part, on the damage, andadjusting the set of stimulation parameters to maximize the amount ofhydrocarbon produced during hydrocarbon production.
 2. The method ofclaim 1, where at least one stimulation parameter of the set ofstimulation parameters comprises diversion parameters.
 3. The method ofclaim 1, where at least one stimulation parameter of the set ofstimulation parameters comprises acidizing parameters.
 4. The method ofclaim 3, wherein the acidizing parameters comprise one or moreparameters selected from the group consisting of an acid strength, anacid concentration, an acid volume, and an acid injection rate.
 5. Themethod of claim 1, wherein determining one or more mechanical parametersof the formation comprises determining one or more mechanical parametersselected from the group consisting of Young's modulus, bulk modulus,shear modulus, cohesion, internal friction angle, and pore-collapsepressure.
 6. The method of claim 1, wherein predicting damage to theformation during hydrocarbon production comprises determining one ormore of shear failure and compressive failure within the formation. 7.The method of claim 6, wherein the one or more of shear failure andcompressive failure within the formation is determined based on porosityof the formation.
 8. The method of claim 6, wherein predicting damage tothe formation during hydrocarbon production comprises determining one ormore of shear failure and compressive failure that extends beyond awormhole penetration radius from the wellbore.
 9. The method of claim 1,wherein predicting an amount of hydrocarbon produced during hydrocarbonproduction comprises predicting an amount of hydrocarbon produced duringa time period following providing the optimized stimulation treatment tothe formation.
 10. The method of claim 9, wherein the time period is atleast six months or more.
 11. The method of claim 9, wherein the timeperiod is at least eighteen months or more.
 12. The method of claim 1,wherein predicting an amount of hydrocarbon produced during hydrocarbonproduction comprises predicting an amount of hydrocarbon produced duringthe formation's productive lifetime.
 13. A non-transitory computerreadable medium having instructions stored therein, which when executedby a computer cause the computer to perform operations comprising: foran initial set of stimulation parameters, determining one or moremechanical parameters indicative of weakening of the formation withinthe region of the wellbore caused by the stimulation parameters,predicting damage to the formation due to formation stress duringhydrocarbon production based on the determined mechanical parameters,predicting an amount of hydrocarbon produced during hydrocarbonproduction based, at least in part, on the predicted damage, andadjusting the set of stimulation parameters to maximize the predictedamount of hydrocarbon produced during hydrocarbon production.
 14. Thenon-transitory computer readable medium of claim 1, wherein thestimulation parameters comprise one or more diversion parameters. 15.The non-transitory computer readable medium of claim 1, wherein thestimulation parameters comprise one or more acidizing parameters. 16.The non-transitory computer readable medium of claim 13, whereindetermining one or more mechanical parameters of the formation comprisesdetermining one or more mechanical parameters selected from the groupconsisting of Young's modulus, bulk modulus, shear modulus, cohesion,internal friction angle, and pore-collapse pressure.
 17. Thenon-transitory computer readable medium of claim 13, wherein predictingan amount of hydrocarbon produced during hydrocarbon productioncomprises predicting an amount of hydrocarbon produced during a timeperiod following providing the optimized stimulation treatment to theformation.
 18. The non-transitory computer readable medium of claim 17,wherein the time period is at least six months or more.
 19. Thenon-transitory computer readable medium of claim 17, wherein the timeperiod is at least eighteen months or more.
 20. The non-transitorycomputer readable medium of claim 13, wherein predicting an amount ofhydrocarbon produced during hydrocarbon production comprises predictingan amount of hydrocarbon produced during the formation's productivelifetime.